With verbs that take another verb as an argument (in particular modal verbs) and that imply opportunity, possibility, necessity or similar, it is possible to use the subjunctive II to indicate that what is said to be possible, necessary or similar did indeed not happen (or does/will not happen).
Sie durfte den Kuchen essen.
Sie hätte den Kuchen essen dürfen.
Both sentences state that she was allowed to eat the cake. The only difference is that the latter one implies that she did not eat the cake, while the former one does not imply anything in this regard.
Für einen rechtzeitige Zustellung muss er das Paket morgen abschicken.
Für einen rechtzeitige Zustellung müsste er das Paket morgen abschicken.
Both sentences state that for a timely delivery, he has to send the parcel tomorrow. The second one implicates that the speaker considers it impossible (or very unlikely) that he will do so.
This question contains another example.
Note that this is different from the regular use of the subjunctive II to indicate that something is not real (irrealis), where the verb corresponding to what is irreal is put into the subjunctive II mood. Here, the verb corresponding to what is irreal is an infinitive and thus cannot be put into the subjunctive mood. Instead a conjugated verb modifying the statement (dürfen, müssen) is put into the subjunctive II mood, which does not indicate, however, that what is described by this verb is irreal – the opportunities, possibilities and necessities are real.
Also note that the modal verbs können, müssen, sollen and dürfen (to all of which the above applies) can also be used to indicate that something is possible or likely for which they either have to be used in the subjunctive II mood (dürfen) or the choice between indicative and subjunctive II at most indicates different probabilities (können, müssen, sollen). Thus, without further context and including the regular use of the subjunctive II, the following sentence can mean three different things:
Er müsste das Paket morgen abschicken.
- There is a necessity for him to send the parcel tomorrow, but he will not do it.
- It is likely that he sends the parcel tomorrow.
- With some irreal condition, such as “Wenn ihm die Angelegenheit wichtig wäre”: (If the condition were true) there would be a necessity for him to send the parcel tomorrow.
Now to your example:
[…] sein teurer Anzug müsste dringend in die Reinigung.
This construction deviates a little from the above as there is no other verb modified by müssen, but if you so wish, there is a verb omitted here that indicates what the suit (Anzug) is supposed to do at the dry-cleaner’s (Reinigung), e.g.:
[…] sein teurer Anzug müsste dringend in die Reinigung gegeben werden.
As with the above examples, the only difference to not using the indicative mood (muss) is that the author does not expect that the suit will actually see a dry-cleaner. For example, it would not make sense to use the subjunctive in the following context.
Nach dem Tomatensaucenvorfall muss sein teurer Anzug in die Reinigung. Deshalb wird er ihn Montag vor der Arbeit dort vorbeibringen.
(After the tomato-sauce incidident, his expensive suit needs to be taken to the dry-cleaner’s. Therefore he will bring it there before work on Monday.)
Note that müssen cannot be used to indicate possibilty or likelihood here, because for this purpose, it would have to be used as a real modal verb (with some infintive).
The latter example makes clear that the subjunctive does not indicate a wish here (see Barth Zalewski’s answer), as we do wish that the suit is thoroughly cleaned. Moreover, to indicate (unreal) wishes, the subjunctive has to modify the verb that relates to the content of the wish, which does not exist in your example and which certainly is not müssen: You do not wish that the suit has to be cleaned, but you wish that the suit is cleaned. The wish is implied through the verb müssen here, not through the subjunctive.
A similar example where the subjunctive II is used to indicate a wish is:
Würde er den Anzug nur in die Reinigung geben!
(If he would only give his suit to the dry-cleaner’s.)